Directional variation of effective modulus

Suppose that a cubic crystal is subjected to a uniaxial tensile stress Ghki along the crystallographic direction [hkl]. The direction cosines of the oriented line represented by [hkl, with respect to the underlying rectangular a ia 2a 3—coordinate axes, are denoted by A, B and C, respectively in terms of the Miller indices, 

Prom the stress transformation rules consistent with the tensorial character of stress, it follows that the normal and shear stress acting in the coordinate directions are related to CThki by 

This representation can be simplified through the use of trigonometric identities to yield 

The maximum and minimum values of the effective elastic modulus, from among all possible line orientations in a cubic crystal, are found to exist for orientations along the (100) and (111) directions, respectively. The extreme values and their ratio are 

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